
where for (REF ) (res. ()), we used Lemma REF  by setting \(\lambda =\frac{2(\mathrm {M}-1)}{\mathrm {M}}\)  (resp. \(\lambda =\frac{2(-1)}{}\) ) and \(\sigma =0.5\) . We recall that
since in [1]} the loss function is bounded on \([0,1]\) , we set \(\sigma =0.5\)  in Lemma REF . Now, inserting (REF ) and () into (REF ) and (REF ), from (REF ) we conclude that
\(\mathbb {E}_{U\sim \mathcal {Q}}\left(\mathrm {L}_{\mathrm {P}_{T\mathbf {Z}^\mathrm {M}}}(U)\right)-\mathbb {E}_{U\sim \mathcal {Q}} \mathrm {L}_{\mathbf {Z}_{1:}^{\mathrm {M}}}(U)\le \sqrt{\frac{\mathrm {D}\left(\mathcal {Q}||\mathcal {P}\right)+\log \left(\frac{2}{\delta }\right)}{2(-1)} }\hspace{0.0pt}\nonumber \\+\frac{1}{}\sum _{i=1}^{}\sqrt{\frac{\mathrm {D}\left(\mathcal {Q}||\mathcal {P}\right) + \mathbb {E}_{\mathcal {Q}}\left[\mathrm {D}\left(\mathrm {Q}_i||\mathrm {P}\right)\right]+\log \frac{2\mathrm {M}}{\delta }}{2(\mathrm {M}-1)}},\) 
